What does the "make screencast" button do?
The Make Screencast button allows you to save all actions you make in the Blender window to a video file.

From the doc:


  Screencasts will record your actions over time either as a video or sequence of image files. The type and location of the output is determined by the settings in the Output panel of the Render context window.


So the screencast video is saved to wherever you set under the Render tab > Output rollout.  It will also use whatever settings you set there for video output format as well.